Monte-Carlo Tree Search (MCTS) for Computer...

Monte-Carlo Tree Search (MCTS) for Computer Go

Bruno [email protected]

Université Paris Descartes

AOA class

MCTS for Computer Go 2

Outline

● The game of Go: a 9x9 game● The « old » approach (*-2002)● The Monte-Carlo approach (2002-2005)● The MCTS approach (2006-today)● Conclusion


The game of Go


The game of Go

● 4000 years● Originated from China● Developed by Japan (20th century)● Best players in Korea, Japan, China● 19x19: official board size● 9x9: beginners' board size


A 9x9 game● The board has 81 « intersections ». Initially,it

is empty.


A 9x9 game● Black moves first. A « stone » is played on

an intersection.


A 9x9 game● White moves second.


A 9x9 game● Moves alternate between Black and White.


A 9x9 game● Two adjacent stones of the same color

builds a « string » with « liberties ».● 4-adjacency


A 9x9 game● Strings are created.


A 9x9 game● A white stone is in « atari » (one liberty).


A 9x9 game● The white string has five liberties.


A 9x9 game● The black stone is « atari ».


A 9x9 game● White « captures » the black stone.


A 9x9 game● For advised players, the game is over.

– Hu?

– Why?


A 9x9 game● What happens if White contests black

« territory »?


A 9x9 game● White has invaded. Two strings are atari!


A 9x9 game● Black captures !


A 9x9 game● White insists but its string is atari...


A 9x9 game● Black has proved is « territory ».


A 9x9 game● Black may contest white territory too.


A 9x9 terminal position● The game is over for computers.

– Hu?

– Who won ?


A 9x9 game● The game ends when both players pass.● One black (resp. white) point for each black

(resp. white) stone and each black (resp. white) « eye » on the board.

● One black (resp. white) eye = an empty intersection surrounded by black (resp. white) stones.


A 9x9 game● Scoring:

– Black = 44 – White = 37– Komi = 7.5– Score = -0.5

● White wins!


Go ranking: « kyu » and « dan »

Top professional players

Average playersl

Very beginners

Beginners

Strong players

Very strong players

Pro ranking Amateur ranking

9 dan

1 dan

30 kyu

20 kyu

10 kyu

1 kyu1 dan

6 dan

9 dan


Computer Go (old history)● First go program (Lefkovitz 1960)● Zobrist hashing (Zobrist 1969)● Interim2 (Wilcox 1979)● Life and death model (Benson 1988)● Patterns: Goliath (Boon 1990)● Mathematical Go (Berlekamp 1991)● Handtalk (Chen 1995)


The old approach● Evaluation of non terminal positions

– Knowledge-based– Breaking-down of a position into sub-

positions● Fixed-depth global tree search

– Depth = 0 : action with the best value– Depth = 1: action leading to the position

with the best evaluation– Depth > 1: alfa-beta or minmax


The old approachCurrent position

Evaluation of non terminal positions

Terminal positions

Bounded depth Tree search

361

2 or 3

Huhu?


Position evaluation● Break-down

– Whole game (win/loss or score)– Goal-oriented sub-game

● String capture● Connections, dividers, eyes, life and death

● Local searches– Alpha-beta and enhancements– Proof-number search


A 19x19 middle-game position


A possible black break-down


A possible white break-down


Possible local evaluations (1)

Alive and territory

unstable

alive

dead

alive

Not important


Possible local evaluations (2)alive

unstable

unstablealive + big territory

unstable


Position evaluation● Local results

– Obtained with local tree search– Result if white plays first (resp. black)– Combinatorial game theory (Conway)– Switches {a|b}, >,


Position evaluation


Drawbacks (1/2)● The break-down is not unique● Performing a (wrong) local tree search on a

(possibly irrelevant) local position● Misevaluating the size of the local position● Different kinds of local information

– Symbolic (group: dead alive unstable)– Numerical (territory size, reduction,

increase)


Drawbacks (2/2)● Local positions interact● Complicated● Domain-dependent knowledge● Need of human expertise● Difficult to program and maintain● Holes of knowledge● Erratic behaviour


Upsides

● Feasible on 1990's computers● Execution is fast

● Some specific local tree searches are accurate and fast


The old approach


Average playersl

Very beginners

Beginners

Strong players

Very strong players


9 dan

1 dan

30 kyu

20 kyu

10 kyu10 kyu

1 kyu1 dan

6 dan

9 dan

Old approach


End of part one!● Next: the Monte-Carlo approach...


The Monte-Carlo (MC) approach● Games containing chance

– Backgammon (Tesauro 1989)

● Games with hidden information– Bridge (Ginsberg 2001)– Poker (Billings & al. 2002)– Scrabble (Sheppard 2002)


The Monte-Carlo approach● Games with complete information

– A general model (Abramson 1990)

● Simulated annealing Go – (Brügmann 1993)– 2 sequences of moves– « all moves as first » heuristic– Gobble on 9x9


The Monte-Carlo approach● Position evaluation:

Launch N random gamesEvaluation = mean value of outcomes

● Depth-one MC algorithm:For each move m {

Play m on the ref positionLaunch N random gamesMove value (m) = mean value

}


Depth-one Monte-Carlo

... ... ...

...


Progressive pruning● (Billings 2002, Sheppard 2002, Bouzy &

Helmstetter 2003)

Current best move

Second best

Pruned

Still explored


Upper bound● Optimism in face of uncertainty

– Intestim (Kaelbling 1993), – UCB multi-armed bandit (Auer & al 2002)

Current best promising move

Second best promising

Current best proven move


All-moves-as-first heuristic (1/3)

A B C D

rA

B

D

C



Actual simulation

A D

CB

A B C D

r

o

A

C

B

D

A

B

D

C



Actual simulation

Virtual simulation = actual simulation assuming c is played« as first »

A D

CB

A B C D

r

o o

CA

C A

B B

D D

A

B

D

C


The Monte-Carlo approach● Upsides

– Robust evaluation– Global search– Move quality increases with computing power

● Way of playing– Good strategical sense but weak tactically

● Easy to program– Follow the rules of the game– No break-down problem


Monte-Carlo and knowledge● Pseudo-random simulations using Go

knowledge (Bouzy 2003)– Moves played with a probability depending on

specific domain-dependent knowledge● 2 basic concepts

– string capture 3x3 shapes


Monte-Carlo and knowledge

● Results are impressive– MC(random)


Monte-Carlo and knowledge● Pseudo-random player

– 3x3 pattern urgency table with 38 patterns– Few dizains of relevant patterns only– Patterns gathered by

● Human expertise● Reinforcement Learning (Bouzy & Chaslot

2006)● Warning

– p1 better than p2 does not mean MC(p1) better than MC(p2)


Monte-Carlo Tree Search (MCTS)● How to integrate MC and TS ?● UCT = UCB for Trees

– (Kocsis & Szepesvari 2006)– Superposition of UCB (Auer & al 2002)

● MCTS– Selection, expansion, updating (Chaslot & al)

(Coulom 2006)– Simulation (Bouzy 2003) (Wang & Gelly

2006)


MCTS (1/2)while (hasTime) {

playOutTreeBasedGame()expandTree()outcome = playOutRandomGame()updateNodes(outcome)

}then choose the node with...

... the best mean value

... the highest visit number


MCTS (2/2)PlayOutTreeBasedGame() {

node = getNode(position)while (node) {

move=selectMove(node)play(move)node = getNode(position)

}}


UCT move selection● Move selection rule to browse the tree:

move=argmax (s*mean + C*sqrt(log(t)/n)) ● Mean value for exploitation

– s (=+-1): color to move● UCT bias for exploration

– C: constant term set up by experiments– t: number of visits of the parent node – n: number of visits of the current node


Example

● 1 iteration

1

1/1


Example● 2 iterations

0

1/2

0/1



1

2/3

0/11/1


Example

● 4 iterations2/4

0/11/10/1


Example

● 5 iterations3/5

0/11/10/1 1/1



0/11/20/1 1/1

0/1

3/6


Example● 7 iterations 3/7

0/11/20/1 1/2

0/1 0/1



0/12/30/1 1/2

0/1 0/11/1



0/10/1 1/2

0/1 0/11/1

2/4

0/1



5/10

0/12/40/1 2/3

0/1 0/11/10/1 1/1



0/12/40/1 3/4

0/1 0/11/10/1 1/1 1/1



7/12

0/12/40/1 4/5

0/1 1/21/10/1 1/1 1/1

1/1


Example● Clarity

– C = 0● Notice

– with C != 0 a node cannot stay unvisited– min or max rule according to the node depth– not visited children have an infinite mean

● Practice– Mean initialized optimistically


MCTS enhancements● The raw version can be enhanced

– Tuning UCT C value– Outcome = score or win loss info (+1/-1)– Doubling the simulation number– RAVE – Using Go knowledge

● In the tree or in the simulations– Speed-up

● Optimizing, pondering, parallelizing


Assessing an enhancement● Self-play

– The new version vs the reference version– % wins with few hundred games – 9x9 (or 19x19 boards)

● Against differently designed programs– GTP (Go Text Protocol)– CGOS (Computer Go Operating System)

● Competitions


Move selection formula tuning

● Using UCB– Best value for C ?– 60-40%

● Using « UCB-tuned » (Auer & al 2002)– C replaced by min(1/4,variance)– 55-45%


Exploration vs exploitation

● General idea: explore at the beginning and exploit in the end of thinking time

● Diminishing C linearly in the remaining time– (Vermorel & al 2005)– 55-45%

● At the end:– Argmax over the mean value or over the

number of visits ?– 55-45%


Kind of outcome

● 2 kinds of outcomes– Score (S) or win loss information (WLI) ?– Probability of winning or expected score ?– Combining both (S+WLI) (score +45 if win)

● Results – WLI vs S 65-35%– S+WLI vs S 65-35%


Doubling the number of simulations

● N = 100,000

● Results – 2N vs N 60-40%– 4N vs 2N 58-42%


Tree management

● Transposition tables– Tree -> Directed Acyclic Graph (DAG)– Different sequences of moves may lead to

the same position– Interest for MC Go: merge the results– Result: 60-40%

● Keeping the tree from one move to the next– Result: 65-35%


RAVE (1/3)

● Rapid Action Value Estimation– Mogo 2007– Use the AMAF heuristic (Brugmann 1993)– There are « many » virtual sequences that

are transposed from the actually played sequence

● Result: – 70-30%


RAVE (2/3)● AMAF heuristic● Which nodes to update?● Actual

– Sequence ACBD– Nodes

● Virtual– BCAD, ADBC, BDAC– Nodes

B

A B

B

DD

D C

CC

AA


RAVE (3/3)● 3 variables

– Usual mean value Mu– AMAF mean value Mamaf– M = β Mamaf + (1-β) Mu– β = sqrt(k/(k+3N))– K set up experimentally

● M varies from Mamaf

to Mu


Knowledge in the simulations

● High urgency for...– capture/escape 55-45%– 3x3 patterns 60-40%– Proximity rule 60-40%

● Mercy rule– Interrupt the game when the difference of

captured stones is greater than a threshold (Hillis 2006)

– 51-49%


Knowledge in the tree

● Virtual wins for good looking moves● Automatic acquisition of patterns of pro

games (Coulom 2007) (Bouzy & Chaslot 2005)● Matching has a high cost● Progressive widening (Chaslot & al 2008)● Interesting under strong time constraints ● Result: 60-40%


Speeding up the simulations

● Fully random simulations (2007)– 50,000 game/second (Lew 2006)– 20,000 (commonly eared)– 10,000 (my program)

● Pseudo-random– 5,000 (my program in 2007)

● Rough optimization is worthwhile


Pondering

● Think on the opponent time– 55-45%– Possible doubling of thinking time– The move of the opponent may not be

the planned move on which you think– Side effect: play quickly to think on the

opponent time


Summing up the enhancements● MCTS with all enhancements vs raw MCTS

– Exploration and exploitation: 60-40%– Win/loss outcome: 65-35%– Rough optimization of simulations 60-40%– Transposition table 60-40%– RAVE 70-30%– Knowledge in the simulations 70-30%– Knowledge in the tree 60-40%– Pondering 55-45%– Parallelization 70-30%

● Result: 99-1%


Parallelization● Computer Chess: Deep Blue● Multi-core computer

– Symmetric MultiProcessor (SMP)– one thread per processor– shared memory, low latency– mutual exclusion (mutex) mechanism

● Cluster of computers– Message Passing Information (MPI)


Parallelization

while (hasTime) {playOutTreeBasedGame()expandTree()outcome = playOutRandomGame()updateNodes(outcome)

}


Leaf parallelization


Leaf parallelization● (Cazenave Jouandeau 2007)● Easy to program● Drawbacks

– Wait for the longest simulation– When part of the simulation outcomes is a

loss, performing the remaining may not be a relevant strategy.


Root parallelization


Root parallelization

● (Cazenave Jouandeau 2007)● Easy to program● No communication● At completion, merge the trees● 4 MCTS for 1sec > 1 MCTS for 4 sec● Good way for low time settings and a small

number of threads


Tree parallelization – global mutex


Tree parallelization – local mutex


Tree parallelization● One shared tree, several threads● Mutex

– Global: the whole tree has a mutex– Local: each node has a mutex

● « Virtual loss »– Given to a node browsed by a thread– Removed at update stage– Preventing threads from similar simulations


Computer-computer results● Computer Olympiads

19x19 9x9– 2010 Erica, Zen, MFGo MyGoFriend– 2009 Zen, Fuego, Mogo Fuego– 2008 MFGo, Mogo, Leela MFGo– 2007 Mogo, CrazyStone, GNU Go Steenvreter– 2006 GNU Go, Go Intellect, Indigo CrazyStone– 2005 Handtalk, Go Intellect, Aya Go Intellect– 2004 Go Intellect, MFGo, Indigo Go Intellect


Human-computer results● 9x9

– 2009: Mogo won a pro with black– 2009: Fuego won a pro with white

● 19x19:– 2008: Mogo won a pro with 9 stones

Crazy Stone won a pro with 8 stones Crazy Stone won a pro with 7 stones

– 2009: Mogo won a pro with 6 stones


MCTS and the old approach


Average players

Very beginners

Beginners

Strong players

Very strong players


9 dan

1 dan

30 kyu

20 kyu

10 kyu

1 kyu1 dan

6 dan

9 dan

MCTS

Old approach

9x9 go

19x19 go


Computer Go (MC history)

● Monte-Carlo Go (Brugmann 1993)● MCGo devel. (Bouzy & Helmstetter 2003)● MC+knowledge (Bouzy 2003)● UCT (Kocsis & Szepesvari 2006)● Crazy Stone (Coulom 2006)● Mogo (Wang & Gelly 2006)


Conclusion● Monte-Carlo brought a Big

improvement in Computer Go over the last decade!

– No old approach based program anymore!– All go programs are MCTS based!– Professional level on 9x9!– Dan level on 19x19!

● Unbelievable 10 years ago!


Some references

● PhD, MCTS and Go (Chaslot 2010)● PhD, Reinf. Learning and Go (Silver 2010)● PhD, R. Learning: applic. to Go (Gelly 2007)● UCT (Kocsis & Szepesvari 2006)● 1st MCTS go program (Coulom 2006)


Web links

● http://www.grappa.univ-lille3.fr/icga/● http://cgos.boardspace.net/● http://www.gokgs.com/● http://www.lri.fr/~gelly/MoGo.htm● http://remi.coulom.free.fr/CrazyStone/● http://fuego.sourceforge.net/● ...

http://www.grappa.univ-lille3.fr/icga/http://cgos.boardspace.net/http://www.gokgs.com/http://www.lri.fr/~gelly/MoGo.htmhttp://remi.coulom.free.fr/CrazyStone/http://fuego.sourceforge.net/


Thank you for your attention!

Diapo 1Diapo 2Diapo 3Diapo 4Diapo 5Diapo 6Diapo 7Diapo 8Diapo 9Diapo 10Diapo 11Diapo 12Diapo 13Diapo 14Diapo 15Diapo 16Diapo 17Diapo 18Diapo 19Diapo 20Diapo 21Diapo 22Diapo 23Diapo 24Diapo 25Diapo 26Diapo 27Diapo 28Diapo 29Diapo 30Diapo 31Diapo 32Diapo 33Diapo 34Diapo 35Diapo 36Diapo 37Diapo 38Diapo 39Diapo 40Diapo 41Diapo 42Diapo 43Diapo 44Diapo 45Diapo 46Diapo 47Diapo 48Diapo 49Diapo 50Diapo 51Diapo 52Diapo 53Diapo 54Diapo 55Diapo 56Diapo 57Diapo 58Diapo 59Diapo 60Diapo 61Diapo 62Diapo 63Diapo 64Diapo 65Diapo 66Diapo 67Diapo 68Diapo 69Diapo 70Diapo 71Diapo 72Diapo 73Diapo 74Diapo 75Diapo 76Diapo 77Diapo 78Diapo 79Diapo 80Diapo 81Diapo 82Diapo 83Diapo 84Diapo 85Diapo 86Diapo 87Diapo 88Diapo 89Diapo 90Diapo 91Diapo 92Diapo 93Diapo 94Diapo 95Diapo 96Diapo 97Diapo 98Diapo 99Diapo 100Diapo 101Diapo 102Diapo 103

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